Weisz modulus

GP 9[ [R 16]The extent of internal transport limits was analysed for the wide fixed-bed reactor, using experimental data on carbon monoxide conversion and matter and process parameter data for the reactants [78]. The analysis was based on the Weisz modulus and the Anderson criterion for judging possible differences between observed and actual reaction rates. As a result, it was found that the small particles eliminate internal transport limitations. 

For lipase, initial activity corresponds to the amount of protein that was adsorbed. Specific activity is constant at 1 mmoFs gE for this carrier-enzyme system, which compares to 27% of the free enzyme activity. The trypsin system shows a lower specific activity that is only 10% of the free enzyme. The reason for the lower recovered activity of this system is not known. To rule out possible internal diffusion limitations, the Wheeler-Weisz modulus was estimated, assuming a carrier layer thickness of 0.1 mm for all carriers. Using the data of the experiments performed at 150 rpm, one finds ... 

Then the Weisz modulus which only includes observables is13... 

Fig. 82 Catalyst effectiveness factor T]p as a function of the Weisz modulus, A r]p DaM, and the reaction order, m.

Figure 8. Effectiveness factor 7 as a function of the generalized Weisz modulus i/ pn for various reaction orders. Influence of intraparticle diffusion on the effective reaction rate (isothermal, irreversible reaction in a flat plate).

Comparing eqs 56 and 27, and recalling the definition of the effectiveness factor according to cq 40, yields the following simple relationship between the Thiele modulus and the Weisz modulus ... 

A plot of the effectiveness factor from cq 53 against the Weisz modulus 1ppn from cq 58 gives the curve depicted in Fig. 8 for a first order reaction (flat plate). On the basis of this diagram, the effectiveness factor can be determined easily once the effective reaction rate and the effective diffusivity arc known. 

For practical purposes however, eq 60 again suffers from the disadvantage that the Thiele modulus must be specified in order to calculate the catalyst efficiency. Thus, the intrinsic rate constant must be known. In this situation, instead of directly plotting eq 60, it is more convenient to relate the effectiveness factor to the Weisz modulus, calculated from eq 58. For selected values of the Biot number Bim, such a diagram is given in Fig. 9. 

According to eq 71 the temperature of the catalyst pellet can be calculated as a function of the Weisz modulus, for given values of the modified Prater number and the Biot number for mass transport. 

Figure 10. Effectiveness factor ij as a function of the Weisz modulus ji. Combined influence of intraparticle and interphase mass transfer and interphase heat transfer on the effective reaction rate (first order, irreversible reaction in a sphere, Biot number Bim = 100, Arrhenius number y — 20, modified Prater number ( as a parameter).

Again, eq 75 cannot be used immediately to calculate the overall effectiveness factor, since the modulus fi, which is related to the unknown catalyst temperature, can only be determined when the overall efficiency has been specified (see eqs 71 and 72). Therefore, both sides of eq 74 arc multiplied by 2, resulting in an expression which relates the Weisz modulus ift to the modulus . Then, for a given value of fi, the corresponding value of ij/ is calculated, and from ij/ the unknown catalyst temperature 0S (eq 71). This temperature is substituted into eq 72 to obtain the corresponding value of the Thiele modulus <)>. Dividing ij/ by fi finally yields the overall effectiveness factor which is then plotted against i/f. 

This procedure yields the curves depicted in Fig. 10 for fixed values of Bim and y. and the modified Prater number fi" as a parameter. From this figure, it is obvious that for exothermal reactions (fi > 0) and large values of the Weisz modulus, effectiveness factors well above unity may be observed. The reason for this is that the decline of the reactant concentration over the... 

Any of the curves in Fig. 10, which refer to different values of the modified Prater number fi, tend to approach a certain limiting value of the Weisz modulus for which the overall effectiveness factor obviously becomes infinitely small. This limit can be easily determined, bearing in mind that the effective reaction rate can never exceed the maximum interphase mass transfer rate (the maximum rate of reactant supply) which is obtained when the surface concentration approaches zero. To show this, we formulate the following simple mass balance, analogous to eq 62 ... 

The maximum effective reaction rate is obtained for the limiting value of cs = 0. This means that the product of the effectiveness factor and the second Damkohler number can never exceed unity. A comparison of the definition of the Weisz modulus (eq 56) with the definition of Dan (eq 78) gives the equivalence... 

As stated above, instead of plotting the effectiveness factor against the Thiele modulus which contains the unknown, intrinsic rate constant, it is often more convenient to relate it to the observable Weisz modulus i/f. This leads to the representation given in Fig. 14 for the same situation as depicted in Fig. 13. The dashed portions of the curves indicate the regions in which a unique solution of the effectiveness factor docs not exist, corresponding to the regions of multiple solutions in Fig. 13. 

Roberts and Satterfield [87, 88] analyzed this type of reaction. On the basis of numerical calculations for a flat plate, these authors presented a solution in the form of effectiveness factor diagrams, from which the effectiveness factor can be determined as a function of the Weisz modulus as well as an additional parameter Kp s which considers the influence of the different adsorption constants and effective diffusivities of the various species [91], The constant K involved in this parameter is defined as follows ... 

In Fig. 19, calculated curves of the effectiveness factor versus the Weisz modulus are shown for different values of Kpis [91]. For comparison, this diagram also contains the curves corresponding to the results which apply to simple, irreversible power rate laws of zeroth, first and second order. From this figure it is obvious that a strong adsorption of at least one of the products leads to a similar decrease of the effectiveness factor as it is observed in the case of a reversible reaction. 

Beside the convenient representation of the effectiveness factor as a function of the observable Weisz modulus, this diagram has the additional advantage that the error arising from an approximation of the truly hyperbolic form of the rate expression by a simple power rate law with integer order can be estimated. 

Inspection of the curves of the effectiveness factor versus the Weisz modulus for different values of Kp. s and E reveals two interesting phenomena when E > 0 (Fig. 20) [87, 88, 91]. At first, for large values of Ay i.s (10-100) effectiveness factors above unity may occur even though isothermal conditions prevail. This can be explained by the fact that the reaction rate given by eq 103 has a maximum for certain combinations of p and P2. This maximum results from the assumption that the rate is proportional to the concentration of the adsorbed reactants At and A2 which compete for adsorption sites on the active (inner) surface. When, for example, Ai is adsorbed more strongly than A2, then a raised partial pressure of Ai, at constant partial pressure of A2, will lead to a displacement of A2 from the surface, and hence to a lowered reaction rate. By a quantitative analysis, it can be shown that effectiveness factors above unity will appear whenever Kp, is greater than (E + 2)/E [91]. 

The effectiveness factor versus the Weisz modulus according to Kao and Satterfield [61] is shown in Fig. 21 for C = 0.5 and different values of B. From this diagram, a similar behavior is seen as in the case of a simple, first order, reversible reaction (see Fig. 18) with decreasing value of B, the effectiveness factor is reduced. A decline of the effectiveness factor is also observed for a rise of the parameter C, which corresponds to a shift towards the chemical equilibrium, and hence to a reduction of the net reaction rate [91]. 

To compare the change in overall selectivity, eqs 135 and 152 would have to be integrated over a range of conversions. However, the resulting lengthy expressions are not given here. Instead, Fig. 23 shows a plot of the relative point selectivity obtained from eq 153 versus the Weisz modulus... 

While it is the purpose of a study to determine, for example, the intrinsic rate one cannot determine whether this criterion is satisfied. Therefore, the following combination is introduced that yields a procurable quantity (an observable) which is referred to as the Wheeler-Weisz modulus [4, 8]. From series expansion, and since tli is close to 1, the following criterion follows for an nth-order reaction [27] ... 

Figure 12. Isothermal internal effectiveness factor as a function of the Wheeler-Weisz modulus for different reaction orders.

Figure 12 shows the effectiveness factor as a function of the Wheeler-Weisz modulus for different reaction orders, indicating that criterion (33) holds for the generalized Thiele modulus. Due to the definition of L it is fairly independent of the catalyst geometry. 

To check for internal mass transfer limitation, it is possible to use the nondimensional Weisz modulus, Thiele modulus (Levenspiel, 1998) ... 

Here, is the experimental mean rate of reaction per unit volume of catalyst, L is a characteristic length of the porous photocatalyst (i.e., the film thickness), t is the pore tortuosity (taken as three), D is the diffusion coefficient of the pollutant in air, Cg is the mean concentration at the external surface, and e is the catalyst grain porosity (0.5 for Degussa s P25). Such a treatment was performed by Doucet et al. (2006) while taking D of the pollutants to be approximately 10 m s. The estimated Weisz modulus ranged between 10 and 10, depending on the type of pollutant, that is, some three to five orders of magnitude smaller than the value of unity, which is often taken as a criterion for internal mass transport limitation. 

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